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G = C7×A4order 84 = 22·3·7

Direct product of C7 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C7×A4, C22⋊C21, (C2×C14)⋊1C3, SmallGroup(84,10)

Series: Derived Chief Lower central Upper central

C1C22 — C7×A4
C1C22C2×C14 — C7×A4
C22 — C7×A4
C1C7

Generators and relations for C7×A4
 G = < a,b,c,d | a7=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

3C2
4C3
3C14
4C21

Character table of C7×A4

 class 123A3B7A7B7C7D7E7F14A14B14C14D14E14F21A21B21C21D21E21F21G21H21I21J21K21L
 size 1344111111333333444444444444
ρ11111111111111111111111111111    trivial
ρ211ζ3ζ32111111111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ3    linear of order 3
ρ311ζ32ζ3111111111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ32    linear of order 3
ρ41111ζ7ζ75ζ72ζ76ζ73ζ74ζ75ζ76ζ7ζ73ζ72ζ74ζ73ζ75ζ72ζ76ζ73ζ74ζ74ζ7ζ75ζ72ζ76ζ7    linear of order 7
ρ51111ζ76ζ72ζ75ζ7ζ74ζ73ζ72ζ7ζ76ζ74ζ75ζ73ζ74ζ72ζ75ζ7ζ74ζ73ζ73ζ76ζ72ζ75ζ7ζ76    linear of order 7
ρ61111ζ73ζ7ζ76ζ74ζ72ζ75ζ7ζ74ζ73ζ72ζ76ζ75ζ72ζ7ζ76ζ74ζ72ζ75ζ75ζ73ζ7ζ76ζ74ζ73    linear of order 7
ρ71111ζ74ζ76ζ7ζ73ζ75ζ72ζ76ζ73ζ74ζ75ζ7ζ72ζ75ζ76ζ7ζ73ζ75ζ72ζ72ζ74ζ76ζ7ζ73ζ74    linear of order 7
ρ81111ζ72ζ73ζ74ζ75ζ76ζ7ζ73ζ75ζ72ζ76ζ74ζ7ζ76ζ73ζ74ζ75ζ76ζ7ζ7ζ72ζ73ζ74ζ75ζ72    linear of order 7
ρ91111ζ75ζ74ζ73ζ72ζ7ζ76ζ74ζ72ζ75ζ7ζ73ζ76ζ7ζ74ζ73ζ72ζ7ζ76ζ76ζ75ζ74ζ73ζ72ζ75    linear of order 7
ρ1011ζ3ζ32ζ72ζ73ζ74ζ75ζ76ζ7ζ73ζ75ζ72ζ76ζ74ζ7ζ32ζ76ζ3ζ73ζ3ζ74ζ3ζ75ζ3ζ76ζ3ζ7ζ32ζ7ζ32ζ72ζ32ζ73ζ32ζ74ζ32ζ75ζ3ζ72    linear of order 21
ρ1111ζ3ζ32ζ75ζ74ζ73ζ72ζ7ζ76ζ74ζ72ζ75ζ7ζ73ζ76ζ32ζ7ζ3ζ74ζ3ζ73ζ3ζ72ζ3ζ7ζ3ζ76ζ32ζ76ζ32ζ75ζ32ζ74ζ32ζ73ζ32ζ72ζ3ζ75    linear of order 21
ρ1211ζ3ζ32ζ74ζ76ζ7ζ73ζ75ζ72ζ76ζ73ζ74ζ75ζ7ζ72ζ32ζ75ζ3ζ76ζ3ζ7ζ3ζ73ζ3ζ75ζ3ζ72ζ32ζ72ζ32ζ74ζ32ζ76ζ32ζ7ζ32ζ73ζ3ζ74    linear of order 21
ρ1311ζ32ζ3ζ75ζ74ζ73ζ72ζ7ζ76ζ74ζ72ζ75ζ7ζ73ζ76ζ3ζ7ζ32ζ74ζ32ζ73ζ32ζ72ζ32ζ7ζ32ζ76ζ3ζ76ζ3ζ75ζ3ζ74ζ3ζ73ζ3ζ72ζ32ζ75    linear of order 21
ρ1411ζ3ζ32ζ76ζ72ζ75ζ7ζ74ζ73ζ72ζ7ζ76ζ74ζ75ζ73ζ32ζ74ζ3ζ72ζ3ζ75ζ3ζ7ζ3ζ74ζ3ζ73ζ32ζ73ζ32ζ76ζ32ζ72ζ32ζ75ζ32ζ7ζ3ζ76    linear of order 21
ρ1511ζ32ζ3ζ76ζ72ζ75ζ7ζ74ζ73ζ72ζ7ζ76ζ74ζ75ζ73ζ3ζ74ζ32ζ72ζ32ζ75ζ32ζ7ζ32ζ74ζ32ζ73ζ3ζ73ζ3ζ76ζ3ζ72ζ3ζ75ζ3ζ7ζ32ζ76    linear of order 21
ρ1611ζ32ζ3ζ72ζ73ζ74ζ75ζ76ζ7ζ73ζ75ζ72ζ76ζ74ζ7ζ3ζ76ζ32ζ73ζ32ζ74ζ32ζ75ζ32ζ76ζ32ζ7ζ3ζ7ζ3ζ72ζ3ζ73ζ3ζ74ζ3ζ75ζ32ζ72    linear of order 21
ρ1711ζ32ζ3ζ7ζ75ζ72ζ76ζ73ζ74ζ75ζ76ζ7ζ73ζ72ζ74ζ3ζ73ζ32ζ75ζ32ζ72ζ32ζ76ζ32ζ73ζ32ζ74ζ3ζ74ζ3ζ7ζ3ζ75ζ3ζ72ζ3ζ76ζ32ζ7    linear of order 21
ρ1811ζ32ζ3ζ74ζ76ζ7ζ73ζ75ζ72ζ76ζ73ζ74ζ75ζ7ζ72ζ3ζ75ζ32ζ76ζ32ζ7ζ32ζ73ζ32ζ75ζ32ζ72ζ3ζ72ζ3ζ74ζ3ζ76ζ3ζ7ζ3ζ73ζ32ζ74    linear of order 21
ρ1911ζ3ζ32ζ7ζ75ζ72ζ76ζ73ζ74ζ75ζ76ζ7ζ73ζ72ζ74ζ32ζ73ζ3ζ75ζ3ζ72ζ3ζ76ζ3ζ73ζ3ζ74ζ32ζ74ζ32ζ7ζ32ζ75ζ32ζ72ζ32ζ76ζ3ζ7    linear of order 21
ρ2011ζ32ζ3ζ73ζ7ζ76ζ74ζ72ζ75ζ7ζ74ζ73ζ72ζ76ζ75ζ3ζ72ζ32ζ7ζ32ζ76ζ32ζ74ζ32ζ72ζ32ζ75ζ3ζ75ζ3ζ73ζ3ζ7ζ3ζ76ζ3ζ74ζ32ζ73    linear of order 21
ρ2111ζ3ζ32ζ73ζ7ζ76ζ74ζ72ζ75ζ7ζ74ζ73ζ72ζ76ζ75ζ32ζ72ζ3ζ7ζ3ζ76ζ3ζ74ζ3ζ72ζ3ζ75ζ32ζ75ζ32ζ73ζ32ζ7ζ32ζ76ζ32ζ74ζ3ζ73    linear of order 21
ρ223-100333333-1-1-1-1-1-1000000000000    orthogonal lifted from A4
ρ233-1007757276737475767737274000000000000    complex faithful
ρ243-1007672757747372776747573000000000000    complex faithful
ρ253-1007273747576773757276747000000000000    complex faithful
ρ263-1007377674727577473727675000000000000    complex faithful
ρ273-1007574737277674727577376000000000000    complex faithful
ρ283-1007476773757276737475772000000000000    complex faithful

Permutation representations of C7×A4
On 28 points - transitive group 28T17
Generators in S28
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 24)(9 25)(10 26)(11 27)(12 28)(13 22)(14 23)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 20)(9 21)(10 15)(11 16)(12 17)(13 18)(14 19)
(8 20 24)(9 21 25)(10 15 26)(11 16 27)(12 17 28)(13 18 22)(14 19 23)

G:=sub<Sym(28)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,24)(9,25)(10,26)(11,27)(12,28)(13,22)(14,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,20)(9,21)(10,15)(11,16)(12,17)(13,18)(14,19), (8,20,24)(9,21,25)(10,15,26)(11,16,27)(12,17,28)(13,18,22)(14,19,23)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,24)(9,25)(10,26)(11,27)(12,28)(13,22)(14,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,20)(9,21)(10,15)(11,16)(12,17)(13,18)(14,19), (8,20,24)(9,21,25)(10,15,26)(11,16,27)(12,17,28)(13,18,22)(14,19,23) );

G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,24),(9,25),(10,26),(11,27),(12,28),(13,22),(14,23)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,22),(7,23),(8,20),(9,21),(10,15),(11,16),(12,17),(13,18),(14,19)], [(8,20,24),(9,21,25),(10,15,26),(11,16,27),(12,17,28),(13,18,22),(14,19,23)])

G:=TransitiveGroup(28,17);

C7×A4 is a maximal subgroup of   C7⋊S4

Matrix representation of C7×A4 in GL3(𝔽43) generated by

1100
0110
0011
,
424242
001
010
,
010
100
424242
,
010
001
100
G:=sub<GL(3,GF(43))| [11,0,0,0,11,0,0,0,11],[42,0,0,42,0,1,42,1,0],[0,1,42,1,0,42,0,0,42],[0,0,1,1,0,0,0,1,0] >;

C7×A4 in GAP, Magma, Sage, TeX

C_7\times A_4
% in TeX

G:=Group("C7xA4");
// GroupNames label

G:=SmallGroup(84,10);
// by ID

G=gap.SmallGroup(84,10);
# by ID

G:=PCGroup([4,-3,-7,-2,2,506,1011]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of C7×A4 in TeX
Character table of C7×A4 in TeX

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